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Metamath Proof Explorer

Theorem cdleme28a

Description: Lemma for cdleme25b . TODO: FIX COMMENT. (Contributed by NM, 4-Feb-2013)

Ref Expression
Hypotheses cdleme26.b 
|- B = ( Base ` K )
cdleme26.l 
|- .<_ = ( le ` K )
cdleme26.j 
|- .\/ = ( join ` K )
cdleme26.m 
|- ./\ = ( meet ` K )
cdleme26.a 
|- A = ( Atoms ` K )
cdleme26.h 
|- H = ( LHyp ` K )
cdleme27.u 
|- U = ( ( P .\/ Q ) ./\ W )
cdleme27.f 
|- F = ( ( s .\/ U ) ./\ ( Q .\/ ( ( P .\/ s ) ./\ W ) ) )
cdleme27.z 
|- Z = ( ( z .\/ U ) ./\ ( Q .\/ ( ( P .\/ z ) ./\ W ) ) )
cdleme27.n 
|- N = ( ( P .\/ Q ) ./\ ( Z .\/ ( ( s .\/ z ) ./\ W ) ) )
cdleme27.d 
|- D = ( iota_ u e. B A. z e. A ( ( -. z .<_ W /\ -. z .<_ ( P .\/ Q ) ) -> u = N ) )
cdleme27.c 
|- C = if ( s .<_ ( P .\/ Q ) , D , F )
cdleme27.g 
|- G = ( ( t .\/ U ) ./\ ( Q .\/ ( ( P .\/ t ) ./\ W ) ) )
cdleme27.o 
|- O = ( ( P .\/ Q ) ./\ ( Z .\/ ( ( t .\/ z ) ./\ W ) ) )
cdleme27.e 
|- E = ( iota_ u e. B A. z e. A ( ( -. z .<_ W /\ -. z .<_ ( P .\/ Q ) ) -> u = O ) )
cdleme27.y 
|- Y = if ( t .<_ ( P .\/ Q ) , E , G )
cdleme28a.v 
|- V = ( ( s .\/ t ) ./\ ( X ./\ W ) )
Assertion cdleme28a 
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( s e. A /\ -. s .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( s =/= t /\ ( ( s .\/ ( X ./\ W ) ) = X /\ ( t .\/ ( X ./\ W ) ) = X ) /\ ( X e. B /\ -. X .<_ W ) ) ) -> ( C .\/ ( X ./\ W ) ) .<_ ( Y .\/ ( X ./\ W ) ) )

Proof

Step Hyp Ref Expression
1 cdleme26.b 
 |-  B = ( Base ` K )
2 cdleme26.l 
 |-  .<_ = ( le ` K )
3 cdleme26.j 
 |-  .\/ = ( join ` K )
4 cdleme26.m 
 |-  ./\ = ( meet ` K )
5 cdleme26.a 
 |-  A = ( Atoms ` K )
6 cdleme26.h 
 |-  H = ( LHyp ` K )
7 cdleme27.u 
 |-  U = ( ( P .\/ Q ) ./\ W )
8 cdleme27.f 
 |-  F = ( ( s .\/ U ) ./\ ( Q .\/ ( ( P .\/ s ) ./\ W ) ) )
9 cdleme27.z 
 |-  Z = ( ( z .\/ U ) ./\ ( Q .\/ ( ( P .\/ z ) ./\ W ) ) )
10 cdleme27.n 
 |-  N = ( ( P .\/ Q ) ./\ ( Z .\/ ( ( s .\/ z ) ./\ W ) ) )
11 cdleme27.d 
 |-  D = ( iota_ u e. B A. z e. A ( ( -. z .<_ W /\ -. z .<_ ( P .\/ Q ) ) -> u = N ) )
12 cdleme27.c 
 |-  C = if ( s .<_ ( P .\/ Q ) , D , F )
13 cdleme27.g 
 |-  G = ( ( t .\/ U ) ./\ ( Q .\/ ( ( P .\/ t ) ./\ W ) ) )
14 cdleme27.o 
 |-  O = ( ( P .\/ Q ) ./\ ( Z .\/ ( ( t .\/ z ) ./\ W ) ) )
15 cdleme27.e 
 |-  E = ( iota_ u e. B A. z e. A ( ( -. z .<_ W /\ -. z .<_ ( P .\/ Q ) ) -> u = O ) )
16 cdleme27.y 
 |-  Y = if ( t .<_ ( P .\/ Q ) , E , G )
17 cdleme28a.v 
 |-  V = ( ( s .\/ t ) ./\ ( X ./\ W ) )
18 simp11l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( s e. A /\ -. s .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( s =/= t /\ ( ( s .\/ ( X ./\ W ) ) = X /\ ( t .\/ ( X ./\ W ) ) = X ) /\ ( X e. B /\ -. X .<_ W ) ) ) -> K e. HL )
19 18 hllatd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( s e. A /\ -. s .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( s =/= t /\ ( ( s .\/ ( X ./\ W ) ) = X /\ ( t .\/ ( X ./\ W ) ) = X ) /\ ( X e. B /\ -. X .<_ W ) ) ) -> K e. Lat )
20 simp11r 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( s e. A /\ -. s .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( s =/= t /\ ( ( s .\/ ( X ./\ W ) ) = X /\ ( t .\/ ( X ./\ W ) ) = X ) /\ ( X e. B /\ -. X .<_ W ) ) ) -> W e. H )
21 simp12 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( s e. A /\ -. s .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( s =/= t /\ ( ( s .\/ ( X ./\ W ) ) = X /\ ( t .\/ ( X ./\ W ) ) = X ) /\ ( X e. B /\ -. X .<_ W ) ) ) -> ( P e. A /\ -. P .<_ W ) )
22 simp13 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( s e. A /\ -. s .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( s =/= t /\ ( ( s .\/ ( X ./\ W ) ) = X /\ ( t .\/ ( X ./\ W ) ) = X ) /\ ( X e. B /\ -. X .<_ W ) ) ) -> ( Q e. A /\ -. Q .<_ W ) )
23 simp22 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( s e. A /\ -. s .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( s =/= t /\ ( ( s .\/ ( X ./\ W ) ) = X /\ ( t .\/ ( X ./\ W ) ) = X ) /\ ( X e. B /\ -. X .<_ W ) ) ) -> ( s e. A /\ -. s .<_ W ) )
24 simp21 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( s e. A /\ -. s .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( s =/= t /\ ( ( s .\/ ( X ./\ W ) ) = X /\ ( t .\/ ( X ./\ W ) ) = X ) /\ ( X e. B /\ -. X .<_ W ) ) ) -> P =/= Q )
25 1 2 3 4 5 6 7 8 9 10 11 12 cdleme27cl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( s e. A /\ -. s .<_ W ) /\ P =/= Q ) ) -> C e. B )
26 18 20 21 22 23 24 25 syl222anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( s e. A /\ -. s .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( s =/= t /\ ( ( s .\/ ( X ./\ W ) ) = X /\ ( t .\/ ( X ./\ W ) ) = X ) /\ ( X e. B /\ -. X .<_ W ) ) ) -> C e. B )
27 simp23 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( s e. A /\ -. s .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( s =/= t /\ ( ( s .\/ ( X ./\ W ) ) = X /\ ( t .\/ ( X ./\ W ) ) = X ) /\ ( X e. B /\ -. X .<_ W ) ) ) -> ( t e. A /\ -. t .<_ W ) )
28 1 2 3 4 5 6 7 13 9 14 15 16 cdleme27cl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( t e. A /\ -. t .<_ W ) /\ P =/= Q ) ) -> Y e. B )
29 18 20 21 22 27 24 28 syl222anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( s e. A /\ -. s .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( s =/= t /\ ( ( s .\/ ( X ./\ W ) ) = X /\ ( t .\/ ( X ./\ W ) ) = X ) /\ ( X e. B /\ -. X .<_ W ) ) ) -> Y e. B )
30 simp11 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( s e. A /\ -. s .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( s =/= t /\ ( ( s .\/ ( X ./\ W ) ) = X /\ ( t .\/ ( X ./\ W ) ) = X ) /\ ( X e. B /\ -. X .<_ W ) ) ) -> ( K e. HL /\ W e. H ) )
31 30 23 27 3jca 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( s e. A /\ -. s .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( s =/= t /\ ( ( s .\/ ( X ./\ W ) ) = X /\ ( t .\/ ( X ./\ W ) ) = X ) /\ ( X e. B /\ -. X .<_ W ) ) ) -> ( ( K e. HL /\ W e. H ) /\ ( s e. A /\ -. s .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) )
32 simp33 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( s e. A /\ -. s .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( s =/= t /\ ( ( s .\/ ( X ./\ W ) ) = X /\ ( t .\/ ( X ./\ W ) ) = X ) /\ ( X e. B /\ -. X .<_ W ) ) ) -> ( X e. B /\ -. X .<_ W ) )
33 simp31 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( s e. A /\ -. s .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( s =/= t /\ ( ( s .\/ ( X ./\ W ) ) = X /\ ( t .\/ ( X ./\ W ) ) = X ) /\ ( X e. B /\ -. X .<_ W ) ) ) -> s =/= t )
34 simp32l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( s e. A /\ -. s .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( s =/= t /\ ( ( s .\/ ( X ./\ W ) ) = X /\ ( t .\/ ( X ./\ W ) ) = X ) /\ ( X e. B /\ -. X .<_ W ) ) ) -> ( s .\/ ( X ./\ W ) ) = X )
35 simp32r 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( s e. A /\ -. s .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( s =/= t /\ ( ( s .\/ ( X ./\ W ) ) = X /\ ( t .\/ ( X ./\ W ) ) = X ) /\ ( X e. B /\ -. X .<_ W ) ) ) -> ( t .\/ ( X ./\ W ) ) = X )
36 33 34 35 3jca 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( s e. A /\ -. s .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( s =/= t /\ ( ( s .\/ ( X ./\ W ) ) = X /\ ( t .\/ ( X ./\ W ) ) = X ) /\ ( X e. B /\ -. X .<_ W ) ) ) -> ( s =/= t /\ ( s .\/ ( X ./\ W ) ) = X /\ ( t .\/ ( X ./\ W ) ) = X ) )
37 1 2 3 4 5 6 17 cdleme23b 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( s e. A /\ -. s .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( X e. B /\ -. X .<_ W ) /\ ( s =/= t /\ ( s .\/ ( X ./\ W ) ) = X /\ ( t .\/ ( X ./\ W ) ) = X ) ) -> V e. A )
38 31 32 36 37 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( s e. A /\ -. s .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( s =/= t /\ ( ( s .\/ ( X ./\ W ) ) = X /\ ( t .\/ ( X ./\ W ) ) = X ) /\ ( X e. B /\ -. X .<_ W ) ) ) -> V e. A )
39 1 5 atbase 
 |-  ( V e. A -> V e. B )
40 38 39 syl 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( s e. A /\ -. s .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( s =/= t /\ ( ( s .\/ ( X ./\ W ) ) = X /\ ( t .\/ ( X ./\ W ) ) = X ) /\ ( X e. B /\ -. X .<_ W ) ) ) -> V e. B )
41 1 3 latjcl 
 |-  ( ( K e. Lat /\ Y e. B /\ V e. B ) -> ( Y .\/ V ) e. B )
42 19 29 40 41 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( s e. A /\ -. s .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( s =/= t /\ ( ( s .\/ ( X ./\ W ) ) = X /\ ( t .\/ ( X ./\ W ) ) = X ) /\ ( X e. B /\ -. X .<_ W ) ) ) -> ( Y .\/ V ) e. B )
43 simp33l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( s e. A /\ -. s .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( s =/= t /\ ( ( s .\/ ( X ./\ W ) ) = X /\ ( t .\/ ( X ./\ W ) ) = X ) /\ ( X e. B /\ -. X .<_ W ) ) ) -> X e. B )
44 1 6 lhpbase 
 |-  ( W e. H -> W e. B )
45 20 44 syl 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( s e. A /\ -. s .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( s =/= t /\ ( ( s .\/ ( X ./\ W ) ) = X /\ ( t .\/ ( X ./\ W ) ) = X ) /\ ( X e. B /\ -. X .<_ W ) ) ) -> W e. B )
46 1 4 latmcl 
 |-  ( ( K e. Lat /\ X e. B /\ W e. B ) -> ( X ./\ W ) e. B )
47 19 43 45 46 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( s e. A /\ -. s .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( s =/= t /\ ( ( s .\/ ( X ./\ W ) ) = X /\ ( t .\/ ( X ./\ W ) ) = X ) /\ ( X e. B /\ -. X .<_ W ) ) ) -> ( X ./\ W ) e. B )
48 1 3 latjcl 
 |-  ( ( K e. Lat /\ Y e. B /\ ( X ./\ W ) e. B ) -> ( Y .\/ ( X ./\ W ) ) e. B )
49 19 29 47 48 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( s e. A /\ -. s .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( s =/= t /\ ( ( s .\/ ( X ./\ W ) ) = X /\ ( t .\/ ( X ./\ W ) ) = X ) /\ ( X e. B /\ -. X .<_ W ) ) ) -> ( Y .\/ ( X ./\ W ) ) e. B )
50 1 2 3 4 5 6 17 cdleme23c 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( s e. A /\ -. s .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( X e. B /\ -. X .<_ W ) /\ ( s =/= t /\ ( s .\/ ( X ./\ W ) ) = X /\ ( t .\/ ( X ./\ W ) ) = X ) ) -> s .<_ ( t .\/ V ) )
51 31 32 36 50 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( s e. A /\ -. s .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( s =/= t /\ ( ( s .\/ ( X ./\ W ) ) = X /\ ( t .\/ ( X ./\ W ) ) = X ) /\ ( X e. B /\ -. X .<_ W ) ) ) -> s .<_ ( t .\/ V ) )
52 33 51 jca 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( s e. A /\ -. s .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( s =/= t /\ ( ( s .\/ ( X ./\ W ) ) = X /\ ( t .\/ ( X ./\ W ) ) = X ) /\ ( X e. B /\ -. X .<_ W ) ) ) -> ( s =/= t /\ s .<_ ( t .\/ V ) ) )
53 1 2 3 4 5 6 17 cdleme23a 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( s e. A /\ -. s .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( X e. B /\ -. X .<_ W ) /\ ( s =/= t /\ ( s .\/ ( X ./\ W ) ) = X /\ ( t .\/ ( X ./\ W ) ) = X ) ) -> V .<_ W )
54 31 32 36 53 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( s e. A /\ -. s .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( s =/= t /\ ( ( s .\/ ( X ./\ W ) ) = X /\ ( t .\/ ( X ./\ W ) ) = X ) /\ ( X e. B /\ -. X .<_ W ) ) ) -> V .<_ W )
55 38 54 jca 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( s e. A /\ -. s .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( s =/= t /\ ( ( s .\/ ( X ./\ W ) ) = X /\ ( t .\/ ( X ./\ W ) ) = X ) /\ ( X e. B /\ -. X .<_ W ) ) ) -> ( V e. A /\ V .<_ W ) )
56 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 cdleme27a 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ P =/= Q /\ ( s e. A /\ -. s .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( s =/= t /\ s .<_ ( t .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> C .<_ ( Y .\/ V ) )
57 30 24 23 21 22 27 52 55 56 syl332anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( s e. A /\ -. s .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( s =/= t /\ ( ( s .\/ ( X ./\ W ) ) = X /\ ( t .\/ ( X ./\ W ) ) = X ) /\ ( X e. B /\ -. X .<_ W ) ) ) -> C .<_ ( Y .\/ V ) )
58 simp22l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( s e. A /\ -. s .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( s =/= t /\ ( ( s .\/ ( X ./\ W ) ) = X /\ ( t .\/ ( X ./\ W ) ) = X ) /\ ( X e. B /\ -. X .<_ W ) ) ) -> s e. A )
59 simp23l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( s e. A /\ -. s .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( s =/= t /\ ( ( s .\/ ( X ./\ W ) ) = X /\ ( t .\/ ( X ./\ W ) ) = X ) /\ ( X e. B /\ -. X .<_ W ) ) ) -> t e. A )
60 1 3 5 hlatjcl 
 |-  ( ( K e. HL /\ s e. A /\ t e. A ) -> ( s .\/ t ) e. B )
61 18 58 59 60 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( s e. A /\ -. s .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( s =/= t /\ ( ( s .\/ ( X ./\ W ) ) = X /\ ( t .\/ ( X ./\ W ) ) = X ) /\ ( X e. B /\ -. X .<_ W ) ) ) -> ( s .\/ t ) e. B )
62 1 2 4 latmle2 
 |-  ( ( K e. Lat /\ ( s .\/ t ) e. B /\ ( X ./\ W ) e. B ) -> ( ( s .\/ t ) ./\ ( X ./\ W ) ) .<_ ( X ./\ W ) )
63 19 61 47 62 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( s e. A /\ -. s .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( s =/= t /\ ( ( s .\/ ( X ./\ W ) ) = X /\ ( t .\/ ( X ./\ W ) ) = X ) /\ ( X e. B /\ -. X .<_ W ) ) ) -> ( ( s .\/ t ) ./\ ( X ./\ W ) ) .<_ ( X ./\ W ) )
64 17 63 eqbrtrid 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( s e. A /\ -. s .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( s =/= t /\ ( ( s .\/ ( X ./\ W ) ) = X /\ ( t .\/ ( X ./\ W ) ) = X ) /\ ( X e. B /\ -. X .<_ W ) ) ) -> V .<_ ( X ./\ W ) )
65 1 2 3 latjlej2 
 |-  ( ( K e. Lat /\ ( V e. B /\ ( X ./\ W ) e. B /\ Y e. B ) ) -> ( V .<_ ( X ./\ W ) -> ( Y .\/ V ) .<_ ( Y .\/ ( X ./\ W ) ) ) )
66 19 40 47 29 65 syl13anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( s e. A /\ -. s .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( s =/= t /\ ( ( s .\/ ( X ./\ W ) ) = X /\ ( t .\/ ( X ./\ W ) ) = X ) /\ ( X e. B /\ -. X .<_ W ) ) ) -> ( V .<_ ( X ./\ W ) -> ( Y .\/ V ) .<_ ( Y .\/ ( X ./\ W ) ) ) )
67 64 66 mpd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( s e. A /\ -. s .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( s =/= t /\ ( ( s .\/ ( X ./\ W ) ) = X /\ ( t .\/ ( X ./\ W ) ) = X ) /\ ( X e. B /\ -. X .<_ W ) ) ) -> ( Y .\/ V ) .<_ ( Y .\/ ( X ./\ W ) ) )
68 1 2 19 26 42 49 57 67 lattrd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( s e. A /\ -. s .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( s =/= t /\ ( ( s .\/ ( X ./\ W ) ) = X /\ ( t .\/ ( X ./\ W ) ) = X ) /\ ( X e. B /\ -. X .<_ W ) ) ) -> C .<_ ( Y .\/ ( X ./\ W ) ) )
69 1 2 3 latlej2 
 |-  ( ( K e. Lat /\ Y e. B /\ ( X ./\ W ) e. B ) -> ( X ./\ W ) .<_ ( Y .\/ ( X ./\ W ) ) )
70 19 29 47 69 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( s e. A /\ -. s .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( s =/= t /\ ( ( s .\/ ( X ./\ W ) ) = X /\ ( t .\/ ( X ./\ W ) ) = X ) /\ ( X e. B /\ -. X .<_ W ) ) ) -> ( X ./\ W ) .<_ ( Y .\/ ( X ./\ W ) ) )
71 1 2 3 latjle12 
 |-  ( ( K e. Lat /\ ( C e. B /\ ( X ./\ W ) e. B /\ ( Y .\/ ( X ./\ W ) ) e. B ) ) -> ( ( C .<_ ( Y .\/ ( X ./\ W ) ) /\ ( X ./\ W ) .<_ ( Y .\/ ( X ./\ W ) ) ) <-> ( C .\/ ( X ./\ W ) ) .<_ ( Y .\/ ( X ./\ W ) ) ) )
72 19 26 47 49 71 syl13anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( s e. A /\ -. s .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( s =/= t /\ ( ( s .\/ ( X ./\ W ) ) = X /\ ( t .\/ ( X ./\ W ) ) = X ) /\ ( X e. B /\ -. X .<_ W ) ) ) -> ( ( C .<_ ( Y .\/ ( X ./\ W ) ) /\ ( X ./\ W ) .<_ ( Y .\/ ( X ./\ W ) ) ) <-> ( C .\/ ( X ./\ W ) ) .<_ ( Y .\/ ( X ./\ W ) ) ) )
73 68 70 72 mpbi2and 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( s e. A /\ -. s .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( s =/= t /\ ( ( s .\/ ( X ./\ W ) ) = X /\ ( t .\/ ( X ./\ W ) ) = X ) /\ ( X e. B /\ -. X .<_ W ) ) ) -> ( C .\/ ( X ./\ W ) ) .<_ ( Y .\/ ( X ./\ W ) ) )